Scientific

The recent striking developments in artificial intelligence underscore the idea that we need to (a) understand better what goes on inside neural networks, (b) thereby gain more control over them and (c) develop methods that are less expensive and apply directly to specialized situations. In this talk, I will talk about work on methods using topological methods to address these three problems. In particular, I will include numerous examples.

Transfer systems are combinatorial objects that encode information about equivariant operations. More precisely, a transfer system encodes the transfers (or wrong-way maps) carried by algebras over certain equivariant operads. Thus, transfer systems allow us to use combinatorial tools to study equivariant homotopy theory. Compatible pairs of transfer systems, which are a pair of transfer systems satisfying certain conditions, correspond to multiplicative structures compatible with an underlying additive structure. In particular, compatible pairs are closely related to $N_\infty$-operads which encode commutative structures in equivariant homotopy theory.

In this talk we introduce transfer systems, compatible pairs, and discuss how Rubin functors significantly aid our study of said transfer systems and their pairs. The work discussed in this talk is from two separate projects, the first is joint with DeMark, Hill, Kamel, Niu, Stoeckl, and Yan, and the second is joint with Darnall, Klanderman, Lewis, Shibata, and Trimble.

This is the second of a three lecture course on model categories from the Summer School on Homotopy Colimits, held at the University of Regina.

• Lecture 1
• Lecture 2
• Lecture 3

This is the third of a three lecture course on equivariant homotopy theory from the Summer School on Homotopy Colimits, held at the University of Regina.

• Lecture 1
• Lecture 2
• Lecture 3

This is the first of a three lecture course on equivariant homotopy theory from the Summer School on Homotopy Colimits, held at the University of Regina.

• Lecture 1
• Lecture 2
• Lecture 3